Focused geometry tournament v2: five hypotheses, three observation regimes
This tranche deliberately narrows the earlier 46 observable classes to five physically economical families: ordinary/null, constant tilt, finite thickness, constant displacement and smooth bend. The ordering is a working priority based on parsimony, compatibility with current evidence and reader feasibility. It is not a posterior probability ranking.
The executable algebra is
This matters because five semantic hypotheses do not necessarily give five empirical alternatives. The tournament evaluates three declared regimes.
| Regime | Observable classes | Result |
|---|---|---|
| current-relative | 2 | ordinary, tilt, thickness and displacement alias; bend retains only a nonlinear directional remainder |
| calibrated-relative | 4 | displacement still aliases ordinary; tilt, thickness and bend separate |
| anchored-full | 5 | all five families separate |
The current-relative quotient includes unknown directional zero points, width and coherence zero points, and an ordinary perspective-expansion field. Under that quotient, a constant directional signal is calibration, a constant width signal is calibration, and a constant displacement is not observed. Smooth bend is not completely erased because normalized directional mixing has a nonlinear spatial remainder. The earlier eight-cluster result nevertheless holds it: flat generalized better than bend.
The calibrated-relative regime fixes directional and width zero levels but still lacks a reader of absolute position in the extra coordinate. It can test tilt, thickness and bend, but it must abstain between ordinary and constant displacement.
The anchored-full regime adds an absolute displacement-sensitive channel. Only there do all five semantic families become five observable classes. Finite-difference Jacobians confirm local parameter rank for every non-null family in this regime.
Each regime is exercised with deterministic nuisance injection and Gaussian noise. Classification is scored at the observable-class level, not the semantic-label level, so equivalence correctly produces abstention rather than a fabricated decision. The evidence record contains the class memberships, local ranks, trial counts, abstention counts and confusion registers.
The first unpenalized current-relative run also exposed an expected failure mode: flexible bend candidates fit noise better than the null class. The sealed tournament therefore reports both unpenalized recovery and recovery after a declared one-degree-of-freedom likelihood-ratio penalty of 6.63. This control is part of the evidence rather than an invisible tuning step. It reduces false bend calls while also recording the corresponding missed weak bends.
The next measurement priority is therefore calibration, not a larger hypothesis language: first establish directional and width zero levels; add an absolute anchor only if testing constant displacement justifies its cost. No result in this tranche promotes a real-data claim.